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What is semiholonomic?

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What is semiholonomic is? Holonomic actually means two point can be expressed by a equation. Non-holonomic can be expressed but, not properly.

Holonomic e.g. $r^2-a^2=0$

Non-holonomic e.g. $r^2-a^2\ge 0$

So, how can I express semiholonomic?

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Semi-holonomic constraints look something like the following: $$f(\mathbf{q},t)=\sum_{i=1}^nf_i(\mathbf{q},t)\dot{q}_i+f_0(\mathbf{q},t)=0$$ with the requirement that $f(\mathbf{q},t)$ be integrable. This expression should look a lot like the total time derivative of some function $F(\mathbf{q},t)$, if we assume that $f_i(\mathbf{q},t)$ and $f_0(\mathbf{q},t)$ are partial derivatives of $F(\mathbf{q},t)$ with respect to $q_i$ and $t$, respectively. The presence of the factors of $\dot{q}_j$ mean that our constraint appears to be, strictly speaking, non-holonomic, but the integrability means that you could also view it as a holonomic constraint in disguise. This speaks in part to the fact that "non-holonomic" has varying definitions depending on who you ask, including:

  • Any constraint that is simply not holonomic (which includes semi-holonomic constraints).
  • A constraint that is not integrable (which excludes semi-holonomic constraints).
  • A constraint of the form $$g(\mathbf{q},\dot{\mathbf{q}},t)=0$$ that is also not holonomic (which includes semi-holonomic constraint but excludes the "non-holonomic" example you give, as it contains an inequality).
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